Preservice Teachers Experiencing Mathematics through Moon Projects and Spinning Tops
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During this past year, we created an inquiry-based environment within our mathematics methods courses through the design of activities and projects, which allowed preservice teachers to investigate, analyze, and communicate “investigable” realms of physical and mathematical phenomena. Our goal was to allow preservice teachers to experience such instruction and to see the value of learning in a project-enhanced environment, so they would be encouraged to design this kind of instruction in their future classrooms.
Current research indicates that inquiry and higher-order thinking are directly connected (Kelly, 1999). Bishop (1988) indicated that inquiry is especially significant for mathematics education, given the dominant misconception that mathematics is viewed as a collection of facts and procedures and that most mathematics classroom are “techniques-oriented.” Richards (1991) offered a notion of “inquiry by design” and argued that “students will not become active learners by accident, but by design, through the use of the plans that we structure to guide exploration and inquiry” (p. 38).
Borasi and Siegal (1992) defined an inquiry cycle, consisting of problem sensing, problem formation, search, and resolution, as an instructional experience organized to engage students in inquiry. Specifically, this inquiry cycle consists of setting the stage; developing one’s question; identifying approaches, resources, and questioning tools; conducting research; collaborating with others; communicating with outside audiences; identifying problems; and inviting new beginnings (p.380).
Preservice teachers need to experience the “practice” of inquiry-based learning through full participation during investigations in methods courses and by discussing their applications with K-12 students. Most preservice teachers have not had personal experiences with inquiry-based learning during their K-12+ educations, so they lack the background to implement this kind of instruction.
Study and Methods
Our research occurred within mathematics methods classes over a period of four semesters. Students conducted an initial moon investigations assignment with a follow-up moon project and a spinning top investigation. All activities required students to keep journals and to document their thinking processes through narrative.
We pursued the following questions:
oHow will providing students with open-ended observations and writing tasks assist them in making sense of physical phenomena?
oWhat will students learn from the inquiry investigations, and will they plan such investigations within their own classrooms?
oWhat mathematical content emerges through these investigations, and what do students’ conjecture?
Data collection included surveys, interviews, journals, and videotaped lessons. Student records documented their evolution of thought. Final interviews and summative narratives determined students’ self-perceptions of which project investigations were most conducive in aiding their understanding.
Our initial assignment asked students to conduct daily moon observations for five weeks. They were required to make sense of their moon observations by writing whatever they wished or thought was relevant. Much mathematics can and was considered when trying to understand the cause of the moon phases. Mathematics included geometrical configurations of the moon, Sun, Earth system; examining the elliptical orbital paths of the moon about the Earth and the moon and Earth about the sun; measuring in degrees the location of the moon in the sky relative to the horizon or relative to other objects in the sky; discovering the differences between sidereal time (the amount of time it takes the moon to orbit the Earth and appear at the same place on the Celestial Sphere) and the lunar month (the amount of time between successive new moons, or full moons, or any two successive, similar phases); possible relationships that occur between the angular displacement of the moon from the Sun and the percent of the moon lit.
Prior to the students’ follow-up moon projects, but after the initial moon assignment, students conducted an investigation with tops, in which tops were used to scaffold them towards success with their moon projects. The assignment began with brainstorming about specific features of a “successful” top. Most groups listed criteria such as, spins:
1) at least five seconds, 2) perpendicular to the floor, 3) on a point, and 4) with a funnel or cone shape.
Student groups were asked to create their top. With a variety of materials available, students created, tested, revised, and collected data with their tops. After initial explorations, groups formulated conjectures.
Some mathematical questions involved a top’s geometry, ratios of axis length below the top’s body to the axis length above, varied dowel rod lengths while holding the top’s diameter constant, and location of weight distribution (Figure 2).
Figure 1 – Preservice teachers test their conjecture that a successful top must have sides of equal length.
Table 1 – Preservice teachers explore the effect of axis length on spin time of top and investigate how the spin time is affected by the ratio of dowel length below top to dowel length above top.
Plate’s Diameter—18.4 cm
Total axis length—8.7cm
Cm. below plate Cm. above plate Ratio Spin time
1.2 cm 7.1 cm 0.17 25.00 sec
1.7 cm 6.4 cm 0.27 19.12 sec
2.3 cm 5.9 cm 0.39 14.20 sec
Total axis length—18.6 cm
Cm. below plate Cm. above plate Ratio Spin time
2.7 cm 15.9 cm 0.17 8.80 sec
3.95 cm 14.65 cm 0.27 2.80 sec
5.2 cm 13.4 cm 0.39 2.50 sec
Figure 2 – Preservice teachers discover that their most successful top’s body was located one inch from the floor; its weight was equally distributed along the top’s rim.
Figure 1 displays a group’s conjecture that for a top to be “successful,” it must have equal length sides. They found this untrue since right triangular and rectangular shaped tops were just as good at spinning as their equal-lengthed counterparts.
Table 1 illustrates two interesting features. Firstly, the best spin time was obtained for the top with an axis length equal to the top’s radius. The second interesting relationship involved the ratio of the axis length below the spinning plate to the axis length above. The ratio that produced the best spin time was 0.17, which occurred when using both an axis length of 8.7 cm and an axis length of 18.6 cm. This group concluded that further data collection would be needed to test both emerging mathematical relationships. Figure 2 displays a bar graph where this group discovered that their most successful top’s body was located one inch from the floor with weights evenly distributed along the top’s rim.
Most found the tops investigation accessible and interesting in that it contextualized mathematical concepts, such as ratios, limits, statistics, and geometrical relationships.
Emerging questions from the initial moon assignment were incorporated into follow-up projects. Investigated topics included light pollution, relationships between the rising and setting times of the moon and sun, crime rate during a full moon, and cows’ milk production at various moon’s phases. Investigations were a unique blend of statistical mathematics, scientific data collection, and narrative. The following table displays the mathematics utilized in each investigation in order to answer their research questions.
Follow-up Moon Projects. Ø Mathematics Utilized
How does light pollution affect what we can see? Ø Star counts from city parking lot, from in-town home, and from rural location.Ø Percentage differences in star counts.
What relationships can be found between the rise and set times of the moon and sun? Ø Collected data from the local paper and a naval website on the moon and sun rise and set time.Ø Graphical representation of data to compare the differences in the changes in time.
How does the moon affect crime? Ø Research statistics from Internet.Ø Collected data on number of crimes committed in Lubbock during the year 2002, and specifically the number of crimes committed on full moon days during 2002.Ø Graphical representation of data to compare the number of average crimes per day with the actual number of crimes per full moon day.
Is there a correlation between the full moon and cattle behavior? Ø Research statistics from Internet.Ø Interviews (concerning milk production per day and number of calves born per day) with veterinarians, feed-lot workers, dairy owners, and farmers/ranchers.Ø Graphical representation of data (scatter plots and pie charts).
Student volunteers were interviewed at the end of the term. Students were asked what they considered to be the purpose of the inquiry activities enacted within their methods course.
One student reported the following:
“For inquiry, you can go and figure out what you want to know. It’s like what we did in the moon investigations. There are so many questions that I wanted to know just by doing that little project…By doing the observations, I developed questions on my own. I think by now I will remember more what I have found out by having questions I wanted to know rather than just something you wanted us to know.”
Students displayed an increased ability of communicating what they do know, and questioning what more they need to know by coupling their experiential learning with opportunities to construct new thoughts and questions. They communicated through sketches, narration, tabular representations, and graphical representations. They looked for patterns and relationships and were beginning to discover a “reality connection” between theoretical ideas and the real world.
Most Common Document Word Stems:
moon (54), student (48), inquiri (44), mathemat (41), teacher (32), top (29), investig (26), question (21), pre (20), servic (20), pre-servic (20), data (18), cm (18), spin (17), project (16), length (16), learn (15), educ (15), research (15), use (14), phase (14),
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| PRE-SERVICE TEACHERS EXPERIENCING MATHEMATICS THROUGH MOON PROJECTS AND SPINNING TOPS Sandi Cooper Sally McMillan Jennifer Wilhelm Texas Tech University Texas Tech University Texas Tech University Sandi.Cooper@ttu.edu Sally.Tyler@ttu.edu Jennifer.Wilhelm@ttu.edu This article describes inquiry-based environments we created in our mathematics methods courses where we designed projects that allowed pre-service teachers to explore investigate analyze and communicate "investigable" realms of physical and mathematical phenomena. Our goal was for students to experience the value of learning in an inquiry-enhanced environment so they would
| standards for school mathematics. Reston Va.: NCTM National Research Council. (1996). National science education standards. Washington DC: National Academy Press. Peirce C. S. (1982). The fixation of belief. In H.S. Thayer (Ed.) Pragmatism: The classic writings (61-78). Indianapolis IN: Hackett. (Original work published 1877). Richards J. (1991). Mathematical discussions. In E. von Glasersfeld (Ed.) Radical constructivism in mathematics education (pp. 13-51). Dordrecht The Netherlands: Kluwer. Schifter D. (1998). Learning mathematics for teaching: From a teachers seminar to the classroom.
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